Meaning "to maintain as true" is from 1864; specific sense "to make a claim" (on an insurance company) is from 1897. Claim properly should not stray too far from its true meaning of "to demand recognition of a right."

n.

early 14c., "a demand of a right; right of claiming," from Old French claime "claim, complaint," from clamer (see claim (v.)). Meaning "thing claimed or demanded" is from 1792; specifically "piece of land allotted and taken" (chiefly U.S. and Australia, in reference to mining) is from 1851. Insurance sense is from 1878.